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Mirrors > Home > ILE Home > Th. List > fliftf | Unicode version |
Description: The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
flift.1 | |
flift.2 | |
flift.3 |
Ref | Expression |
---|---|
fliftf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 | |
2 | flift.1 | . . . . . . . . . . 11 | |
3 | flift.2 | . . . . . . . . . . 11 | |
4 | flift.3 | . . . . . . . . . . 11 | |
5 | 2, 3, 4 | fliftel 5662 | . . . . . . . . . 10 |
6 | 5 | exbidv 1781 | . . . . . . . . 9 |
7 | 6 | adantr 274 | . . . . . . . 8 |
8 | rexcom4 2683 | . . . . . . . . 9 | |
9 | elisset 2674 | . . . . . . . . . . . . . 14 | |
10 | 4, 9 | syl 14 | . . . . . . . . . . . . 13 |
11 | 10 | biantrud 302 | . . . . . . . . . . . 12 |
12 | 19.42v 1862 | . . . . . . . . . . . 12 | |
13 | 11, 12 | syl6rbbr 198 | . . . . . . . . . . 11 |
14 | 13 | rexbidva 2411 | . . . . . . . . . 10 |
15 | 14 | adantr 274 | . . . . . . . . 9 |
16 | 8, 15 | syl5bbr 193 | . . . . . . . 8 |
17 | 7, 16 | bitrd 187 | . . . . . . 7 |
18 | 17 | abbidv 2235 | . . . . . 6 |
19 | df-dm 4519 | . . . . . 6 | |
20 | eqid 2117 | . . . . . . 7 | |
21 | 20 | rnmpt 4757 | . . . . . 6 |
22 | 18, 19, 21 | 3eqtr4g 2175 | . . . . 5 |
23 | df-fn 5096 | . . . . 5 | |
24 | 1, 22, 23 | sylanbrc 413 | . . . 4 |
25 | 2, 3, 4 | fliftrel 5661 | . . . . . . 7 |
26 | 25 | adantr 274 | . . . . . 6 |
27 | rnss 4739 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 |
29 | rnxpss 4940 | . . . . 5 | |
30 | 28, 29 | sstrdi 3079 | . . . 4 |
31 | df-f 5097 | . . . 4 | |
32 | 24, 30, 31 | sylanbrc 413 | . . 3 |
33 | 32 | ex 114 | . 2 |
34 | ffun 5245 | . 2 | |
35 | 33, 34 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cab 2103 wrex 2394 wss 3041 cop 3500 class class class wbr 3899 cmpt 3959 cxp 4507 cdm 4509 crn 4510 wfun 5087 wfn 5088 wf 5089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 |
This theorem is referenced by: qliftf 6482 |
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